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Square Roots

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Square Roots

The square root of a number is the number that, when squared (multiplied
by itself), is equal to the given number. For example, the square root of 16,
denoted 16^{1/2} or , is 4, because 4^{2} = 4×4 = 16. The square
root of 121, denoted , is 11, because 11^{2} = 121.
= 5/3, because (5/3)^{2} = 25/9.
= 9, because 9^{2} = 81. To take the square root of a
fraction, take the square root of the numerator and the square root of the
denominator. The square root of a number is always positive.

All perfect squares have square roots that are whole numbers. All fractions
that have a perfect square in both numerator and denominator have square roots
that are rational numbers.
For example, = 9/7. All other positive numbers have
squares that are non-terminating, non-
repeating decimals, or irrational
numbers. For example, = 1.41421356... and = 2.19503572....

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Square Roots of Negative Numbers

Since a positive number multiplied by itself (a positive number) is always
positive, and a negative number multiplied by itself (a negative number) is
always positive, a number squared is always positive. Therefore, we
cannot take the square root of a negative number.

Taking a square root is almost the inverse
operation of taking a square. Squaring a positive
number and then taking the square root of the result does not change the number:
= = 6. However, squaring a
negative number and then taking the square root of the result is equivalent to
taking the opposite of the negative
number: = = 7. Thus, we
conclude that squaring any number and then taking the square root of the
result is equivalent to taking the absolute value of the given number. For example, = | 6| = 6, and
= | - 7| = 7.

Taking the square root first and then squaring the result yields a slightly
different case. When we take the square root of a positive number and then
square the result, the number does not change: ()^{2} = 11^{2} = 121. However, we cannot take the square root of a negative
number and then square the result, for the simple reason that it is impossible
to take the square root of a negative number.

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Cube Roots and Higher Order Roots

A cube root is a number that, when cubed, is equal to the given number.
It is denoted with an exponent of "1/3". For example, the cube root of 27 is
27^{1/3} = 3. The cube root of 125/343 is (125/343)^{1/3} = (125^{1/3})/(343^{1/3}) = 25/7.